Titlebook: Real Analysis: Foundations; Sergei Ovchinnikov Textbook 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to S

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Real Numbers,lutions of simple quadratic equations (cf. Theorem . and Exercise .), to measure the length of a hypotenuse of a right triangle (cf. Theorem .), and to find a limit of an intuitively convergent Cauchy sequence (cf. Example .).

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Continuous Functions,e sets. Then, the classes of connected and compact subsets of an ordered field are defined and their properties are investigated. We show that some properties of those special sets are equivalent to the completeness property, and therefore characterize the field of real numbers .. In conclusion of t

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Infinite Series,” ordered fields convergent series are “finite”, that is, they terminate with zeros. Also, a sufficiency condition for convergence is established for series with terms in a non-Archimedean, Cauchy complete field.

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Continuous Functions,operties of those special sets are equivalent to the completeness property, and therefore characterize the field of real numbers .. In conclusion of this section, we prove the Heine–Borel and Borel–Lebesgue theorems characterizing compact sets of real numbers.

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Infinite Series,” ordered fields convergent series are “finite”, that is, they terminate with zeros. Also, a sufficiency condition for convergence is established for series with terms in a non-Archimedean, Cauchy complete field.

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